Table of contents

From September 15 to 18, 2021, the International Symposium "Nonequilibrium Processes in Continuous Media" (PHD-Symposium 2021) was held. The symposium was devoted to the analysis of current problems in the field of non-equilibrium processes in continuous media and discussion of ways to solve them, exchange of advanced scientific achievements and practical experience, establishment of new and strengthening of existing scientific ties, creation of necessary conditions for stimulating and supporting gifted youth, dissemination and popularization of scientific knowledge and innovative technologies. The languages of the Symposium are English and Russian.

List of Symposium Chairwomen, Symposium participants, Chairwomen, Program Committee, Plenary Lecturers are available in the pdf.

All papers published in this volume have been reviewed through processes administered by the Editors. Reviews were conducted by expert referees to the professional and scientific standards expected of a proceedings journal published by IOP Publishing Publishing.

• Type of peer review: Single Anonymous

• Conference submission management system: Morressier

• Number of submissions received: 54

• Number of submissions sent for review: 50

• Number of submissions accepted: 25

• Acceptance Rate (Submissions Accepted / Submissions Received × 100): 46.3

• Average number of reviews per paper: 2

• Total number of reviewers involved: 14

• Contact person for queries:

Name: Tatyana Lyubimova

Email: lubimova@psu.ru

Affiliation: Department of Theoretical Physics, Perm State University, Perm, Russia

In this paper, we analytically study the mechanism of the onset of the liquid/gas flow in a non-isothermal stratified medium. The flow is induced by small spatial inhomogeneities of the exchange coefficients. In a statically stable stratified medium heated strictly from above, as is known, there is heat diffusion directed from top to bottom. We consider the system, which is slightly non-invariant with respect to translations along with horizontal directions. It may occur, for example, due to the dependence of the thermal conductivity coefficient on horizontal coordinates. In this case, we show that it could result in the rise of horizontal inhomogeneity in the distributions of buoyancy and hydrostatic pressure and, consequently, in the onset of the horizontal advection. We consider harmonic variations of the thermal conductivity of small amplitude. By applying a linear approximation to a set of governing equations, we derive explicit analytical expressions for temperature perturbations and velocity components. Finally, we investigate the possibility of an intense response of the system on slight initial symmetry breaking in a definite range of parameter values.

The numerical simulation of the dynamics of a liquid with solute of high concentration in two horizontal layers of a porous medium with different permeability coefficients is carried out. The dynamics of the solute is modeled within the framework of a nonlinear MIM model that takes into account the adhesion and separation process at the skeleton of a porous medium. The modeling was carried out in a two-dimensional setting. The mobile (moving together with the carrier fluid) and immobile (deposited on the skeleton) phases are modeled by the finite volume method with adaptive mesh refinement algorithm. At each point in space, functions of volumetric saturation are set, and a kinematic equation is solved that describes the dynamics of adhesion and separation. The effective permeability of a porous medium depends on the saturation function of the immobile phase and is calculated using the Kozeny-Karman formula. The obtained numerical data are compared with the results of a linear analysis of stability, and the nonlinear regimes of concentration convection and the distribution of impurities at high supercriticalities are studied. In the case of high ratio of permeability coefficients of upper and lower layers, convection occurs locally in a more permeable sublayer, even with a decrease in its thickness. With an increase in supercriticality the penetration of the convective flow into the poorly permeable sublayer and the transition from local convection to large-scale convection are observed.

A computational package CrystarPack developed by the author is presented, which is designed to solve problems of continuum mechanics according to an implicit scheme using finite difference and finite volume methods on a Cartesian grid with a refined step, as well as a modified iterative Newton method with global convergence. The package has the form of a dll-library and is flexible and easy to customize for various physical problems, allowing its functionality to be included in the user-written code in any of the imperative programming languages.

Evaporation progress of a macroscopic-scale sessile droplet with the pinned triple line on heated isothermal substrates has been experimentally investigated in the terrestrial gravity, in order to study the interface effect, heat and mass transfer behaviours during the phase change process. The experiments were carried out in a closed chamber in which the environment temperature and pressure were regulated. The contact radius, liquid volume and contact angle during evaporating were observed to study the influence of the gravity effect on the drop shape. The instant evaporation rate was calculated and compared with the theoretical prediction to analyse the coupling influence of diffusion and thermal convection. The effect of substrate temperature on the heat flux were also focused on. It was found that the evolution of heat flux density could be separated into four stages, began with the rapid increase and warm up, then switched to the long-time stable stage and ended up with the final rapid decrease stage.

The dynamics of a phase inclusion in a coaxial liquid layer divided with a radial partition is studied experimentally. The working volume of the container is filled with a viscous liquid, inside which an air bubble, immiscible with the main phase, is injected. This inclusion has a lower density than the surrounding liquid does. The container performs rotational oscillations as a whole with the zero average rotation. Such a motion brings to the generation of a harmonically oscillating azimuthal shear flow, which, as a consequence, excites the oscillations of the phase inclusion. During the bubble's oscillations, the displacement of its geometric center follows the sinusoidal law. On the background of such a motion a periodic deformation of the bubble is observed, i.e. the phase boundary starts oscillating. A new and surprising result of the experiments is found, when the light bubble sinks and takes a quasi-steady position near the inner wall of the layer.

The results of a direct numerical simulation are presented which describe the specific process of a porous medium high-temperature siliconizing. The calculations were performed taking into account the diffusion, the weak pressure gradient across a workpiece and the heat release during the condensation of gaseous silicon in pores. Phenomenological formulas for the evaporation and condensation coefficients in dependence on temperature are offered. An evolutionary equation for the temperature distribution in a sample is derived, which takes into account these processes. Mathematical statement of the problem is based on the modified MIM-model with addition Darcy's law. Numerical modeling has been fulfilled by the finite difference method using an explicit scheme. It is shown that the heating release in the sample slows down the process of particle settling. The results of numerical simulation are consistent qualitatively with known experimental data.

The research work shows the method of generating and registration of bubbles which is provided by the finely disperse membrane, the air compressor, the water pump and the optical image registration unit and analyses of images. In order to estimate the size of the fixed bubbles the specialized software on Python programming language was used as a tool of analysis. The software outcomes are presented in the form of diagrams of the distribution of the number of bubbles by the diameter. The results of experiments for solutions of sodium chloride salt and sodium dodecyl sulfate of various concentrations indicated the possibility of controlling both the diameter and the amount of produced air bubbles. One of the main findings of the scientific work is determination of the concentration of sodium chloride, which leads to inhibition of the effect of Sodium dodecyl sulfate on the distribution of bubbles by their size in the experiments.

The effect of rotation velocity modulation on the equilibrium shape of the liquid-air interface in a vertical slot gap rotating around a horizontal axis is studied experimentally. The case is considered when the modulation frequency coincides with the rotation frequency. It is found that in this case, velocity modulation leads to a violation of the axial symmetry of the interface formed by the centrifugal force of inertia. The displacement of the center of symmetry of the interface increases linearly with the amplitude of the velocity modulation and decreases quadratically with the speed of rotation. It is shown that the detected phenomenon is explained by the averaged effect of an external gravitational field on a fluid in a rotating frame of reference.

Some bones of the human skeleton are porous inside and resemble a sponge. The cancellous bone tissue forms the epiphyses of the long bones and practically the entire volume of the cancellous bones. It is known that the structural units of the spongy substance – trabeculae, are located along the lines of action of the greatest mechanical loads created by the muscles attached to the skeleton. If the load during life changes due to some reason, for example, injuries, the characteristics of the profession or lifestyle, then the cancellous bone rebuilds its structure in order to withstand this load. In the presented study, a model is proposed for describing the process of cancellous bone tissue adaptation in the geometric Ilyushin's spaces

A software package for the numerical solution of systems of linear and nonlinear partial differential equations by grid methods is presented, which combines symbolic computations and automatic code generation techniques. Discretization of differential equations is carried out by the methods of finite volumes and finite differences on two-dimensional and three-dimensional regular rectangular grids. The solution of the systems of nonlinear algebraic equations obtained as a result of discretization is performed by Newton's method with employment of third-party SLAE solvers. The generated code can exploit shared and distributed memory parallelisms and take advantage of heterogeneous computations. The results of the numerical solutions of two sample CFD problems of the fluid flow generation in a cylindrical cavity filled with electrically conductive liquid driven the by traveling and rotating magnetic fields are presented.

The paper deals with the problems of stability for a system of circular rings interconnected in such a way that the displacements of these rings, and the rotation angles of their sections at some points, coincide. This has been reduced to a certain variational problem with restrictions on the functions in question in the form of linear equations, and Fourier series are used to obtain a finite-dimensional approximation. The paper also presents the stability problem for a system of circular rings reinforced by inextensible threads that cannot resist compressive forces. In this case, constraints in the form of inequalities arise, and after finite-dimensional approximation the problem reduces to finding the bifurcation points for a nonlinear programming problem in the presence of inequality constraints.

The problem of thermoelastic deformation of a thin-layer coated strip is considered. To consider scale effects, a one-parameter gradient thermoelasticity model is used. The solution to the problem is based on the application of the Fourier transform in the longitudinal coordinate. The inversion of transformants is carried out on the basis of Filon's quadrature formula. The stress-strain state of the coated strip was calculated both in gradient setting and in classical setting. The analysis of the influence of the scale parameter on the distribution of displacements and stresses is made.

The shape of the interface between two liquids in a non-uniformly rotating slotted gap with a circular boundary is experimentally investigated. The liquids have different densities and a high viscosity contrast. We have found that the axisymmetric interface becomes unstable with an increase in the amplitude of the modulation of the rotation rate, a quasi-stationary azimuthally periodic relief excites in a threshold manner. The shape of the interface is studied by photoregistration as a function of the rotation rate, amplitude and frequency of the rotation rate modulation. The relative volume of a viscous liquid and the properties of liquids (viscosity contrast, relative density, coefficient of interfacial tension) vary. We found that the relief develops at the interface regardless of where the more viscous liquid is located, outside or inside.

The in-line smart pig for gas pipeline diagnostics is driven by the pumped gas and performs quality diagnostics only if its speed is constant, when the information reading is uniform along the length of the pipeline. In a low-pressure gas pipeline, if there are inhomogeneities on its walls, the condition of the pig speed constancy is easily broken, as the force of resistance to the pig movement on the stoppers is comparable with the difference of pressure forces acting on its ends. A central bypass channel of constant cross-section is usually used to control the pig velocity. The paper shows insufficiency of such method of regulation, leading to noticeable flow pulsations at the outlet of the channel. The use of variable cross-section of the bypass channel along its length in the form of a Laval nozzle to reduce flow pulsations is proposed. The numerical simulation of gas flow in Laval nozzles of different configuration is performed; it is shown that for all considered nozzles, the pressure pulsation at the channel outlet is 20-30 times less than that in the cylindrical channel, which simplifies the velocity control of such pig. However, it is not possible to completely avoid detachment of the flow from the walls of the expanding part of the nozzle, so in order to reduce the area of the detachment zone and reduce the flow resistance force, additional peripheral bypass channels connecting the inlet end of the pig with the diffuser section of the central bypass are added. Thus, a 40% reduction of gas flow resistance force through the bypass compared to the cylindrical bypass is obtained.

Heat transfer in a horizontal annular region filled with the viscous fluid is studied experimentally. The external boundary of the working volume is formed by an elastic silicone shell, while the internal one – by a cylindrical copper heater. The walls of the shell are brought into symmetric oscillatory motion by two linear servomotors. The temperature outside the external cylinder is maintained constant by pumping the thermally stabilized liquid. The experiments are conducted with the low heating power that corresponds to relatively low values of the Rayleigh number. This allows maintaining a regime, in which the forced, vibrational convection is the main mechanism of heat transfer. Theoretical analysis of the experimental results shows that the structure of the fluid flows is determined by the competition between the free, thermal convection and the forced one – the steady streaming generated by vibration. At sufficiently high intensity of oscillations, the forced convection dominates and leads to the increase of the heat transfer rate.

The motion of a light spherical body in a rotating tilted cylindrical cavity with liquid is studied experimentally. The rotation rate and angle of inclination of the cuvette, the viscosity of the liquid and the characteristics of the body vary. It is shown that with an increase in the rate of the cell rotation, the velocity of the body's ascent decreases. An increase in the angle of inclination of the cylinder relative to the vertical also leads to a decrease in the ascent rate of the body. A paradoxical behaviour of the body is demonstrated: with an increase in the viscosity of the liquid, the velocity of the body's ascent increases while maintaining other parameters (rotation rate, size, density of the body and the angle of inclination). The results of investigations at various angles of inclination, properties of liquid and the body are generalized on the plane of two control dimensionless parameters. Comparison with theory and with experimental results of other authors in the case of vertical orientation of the cavity demonstrates good agreement. It was found that during ascent, the body performs differential rotation relative to the cavity, the rate of which depends on the axial coordinate of the body.

Free translational oscillations of deformable liquid droplet are considered. It is placed into a vessel filled with another liquid. This droplet has an equilibrium revolution shape. The revolution axis of this form is perpendicular two parallel solid plates – floor and cover of vessel. The arbitrary equilibrium contact angle between the interface and the solid substrate can take arbitrary values in the range 0 and π. A contact line velocity is linearly proportional to contact angle deviation from its initial position. Proportionality coefficients (Hocking's parameters) are individual for any substrate. They characterize individual dissipation coefficient, but whole system's dissipation is proportional to a sum of all coefficients. There are three characteristic natural frequency intervals for the case of small aspect ratio. High frequencies are the capillary wave frequencies at the interface. The middle frequency is the fundamental or main one. Low frequencies correspond to the drop oscillations with a stationary contact angle. The cylindrical droplet has the highest frequency.

As is well known, flows in a free atmosphere outside a narrow equatorial zone are characterized by a geostrophic regime. For several applications, it is of interest to estimate the disturbances of the geostrophic wind caused by the action of sources of heat and/or momentum. One possible example is associated with manifestations of internal gravitational waves. A linear analytical model of the geostrophic flow disturbances of a stratified rotating medium caused by the action of stationary sources of momentum and/or buoyancy is considered. The model makes it possible to estimate explicitly the amplitudes of such disturbances depending on the parameters of the sources and the characteristics of the medium.

Numerical modeling of thermophysical processes in the air cooling channel of the titanium sponge reduction reactor is performed. The cooling channel is an area bounded by a retort filled with liquid magnesium and a wall on which heating elements are mounted. Air flows inside the channel, cooling the retort. A mathematical model is constructed based on unsteady Navier-Stokes equations in axisymmetric formulation using a k −ω SST turbulence model. The model takes into account the radiation heat transfer between the retort and reactor walls. Two variants of thermal boundary conditions are considered. The temperature conditions of the retort wall have been calculated, and the profiles of the heat transfer coefficient along the retort wall for a wide interval of airflow rates have been obtained. It is shown that temperature distributions along the retort are not uniform and strongly depend both on external boundary conditions and on cooling intensity. Heat transfer coefficient distributions from its outer wall for different retort heating conditions are plotted, and an empirical formula for calculating the profile of this coefficient is proposed.

The influence of differences in the hydrochemical characteristics of the waters of merging rivers on the formation of density currents and the behavior of the density jump layer is studied numerically. The problem is solved for the velocity regimes of the confluent Chusovaya and Sylva rivers (Russian Federation), which are characterized by significantly different salinity values. To study the specific features of layered structures, numerical simulation is performed in the framework of a two-dimensional approach. The calculations are carried out for a section of the Chusovaya River with a length of 300 m, located downstream from confluence of the Chusovaya and Sylva rivers. The Chusovskoy water intake of the city of Perm is located on this site. In the winter-spring season, in this area a regime is established in which a stable layer of density and velocity jumps is observed. With the confluence parameters typical of the winter low water season in the area of the Chusovskoy water intake, 8 km below the confluence point, field measurements detected fluctuations in the concentration and flow velocity in the density jump layer. The two-dimensional numerical simulations performed in the present work also showed the presence of velocity and concentration fluctuations in the jump layer. We found that vortex structures are formed in the jump layer, leading to fluctuations in the concentration in time at a given point and to the formation of a wave structure of the concentration field at different times.

We study the approach to calculating the scale and shape parameters of the Weibull droplet-size distribution by means of the skewness, kurtosis and variance, obtained by simulation of atomization in an air-assisted atomizer with different viscosities, densities, flow rates and surface tensions of atomizing liquid. Such approach evades necessity to determine the shift parameter and to use the approximation of distribution obtained from the calculation by the Weibull distribution. The major obstacle to an application of it in processing the data about the atomization of droplets (obtained through the calculation) is the high level of fluctuations of skewness and kurtosis in case of small averaging time.

Continuous-flow microfluidic devices are widely used in microbiology, fine organic synthesis, pharmaceuticals, biomedicine, etc. Most applications require rapid mixing of the fluids that pass through the microfluidic chip. The mechanism of natural diffusion is not always efficient due to limitations on the length of the channel. In this work, we numerically study the efficiency of using various mechanisms of natural convection for the mixing of fluids entering the microfluidic chip. Solutions typically differ in buoyancy and diffusion rates of dissolved components, making them sensitive to gravity-dependent instabilities such as Rayleigh-Taylor convection, double diffusion and diffusion layer convection. We consider a Y-shaped microchannel, which is, on the one hand, the simplest, and, on the other hand, a typical element of a microfluidic network. We assume that two miscible solutions independently flow into a common channel where they come into contact. For each type of instability, we numerically estimated the characteristic channel length, after which complete mixing of the solutions occurs. The simulations were performed in the framework of both 2D and 3D models. Finally, we compare the numerical results with the experimental data obtained recently.

Dispersing of the medium from which the desired component is extracted is still one of the main approaches to liquid extraction. However, despite its high efficiency, this approach is still of empirical nature, since the choice of the main parameters - the average drop diameter and the residence time of the drop in the surrounding liquid (the extractant) - is determined by experiment in relation to the initial concentration of the extracted substance (reagent). The main difficulty with a full-scale theoretical treatment and numerical simulation is a three-dimensional shape of the droplet and, accordingly, the lack of experimental data on the structure of convective flows and the dynamics of reagent concentration fields in the moving droplet. The situation can get worse when the extracted component is a surfactant or reactive against the extractant. The paper presents the results of an experimental study of the structure and evolution of flows and distribution of a reagent (acetic acid) diffusing from a rising cylindrical droplet with an insoluble base component into a chemically active medium (aqueous sodium hydroxide solution). The motion of the droplet is accompanied by the development of Marangoni convection and neutralization reaction on the droplet surface.

The liquid drop's natural translational oscillations are considered. The equilibrium form of this drop is a circular cylinder. Its axis of symmetry is perpendicular to two parallel solid substrates. The properties (wetting, roughness etc.) of these surfaces differ from each other. The drop is in another liquid. The contact angles's changes are linearly proportional to the velocities of both contact lines. The Fourier series form by Laplace's operator eigenfunctions are used for the problem solution. A system of complex equations of eigenvalue problem is solved numerically. The main frequency of the translational mode becomes zero after a critical Hocking's parameter in situation of identical plates. The branching point of a decrement curve agrees with the zero point of a fundamental frequency. This frequency may not be vanishing on nonidentical surfaces of plates.